Optics involves the reflection and refraction of light. Reflection occurs when light bounces off a surface, following the law of reflection where the angle of incidence equals the angle of reflection. Refraction is when light changes speed and direction when passing from one medium to another due to a change in index of refraction. Refraction is described by Snell's law, where the ratio of sines of the incident and refracted angles is equal to the ratio of the indices of refraction. Total internal reflection occurs when light passes from a higher to lower index of refraction beyond the critical angle and is completely reflected rather than refracted.

1. Optics
• Reflection • Prisms
• Diffuse reflection • Rainbows
• Refraction • Plane mirrors
• Index of refraction • Spherical aberration
• Speed of light • Concave and convex mirrors
• Snell’s law • Focal length & radius of curvature
• Geometry problems • Mirror / lens equation
• Critical angle • Convex and concave lenses
• Total internal reflection • Human eye
• Brewster angle • Chromatic aberration
• Fiber optics • Telescopes
• Mirages • Huygens’ principle
• Dispersion • Diffraction

2. Reflection
Most things we see are thanks to reflections, since most objects
don’t produce their own visible light. Much of the light incident
on an object is absorbed but some is reflected. the wavelengths of
the reflected light determine the colors we see. When white light
hits an apple, for instance, primarily red wavelengths are
reflected, while much of the others are absorbed.
A ray of light heading towards an object is called an incident ray.
If it reflects off the object, it is called a reflected ray. A
perpendicular line drawn at any point on a surface is called a
normal (just like with normal force). The angle between the
incident ray and normal is called the angle of incidence, i, and
the angle between the reflected ray and the normal ray is called
the angle of reflection, r. The law of reflection states that the
angle of incidence is always equal to the angle of reflection.

3. Law of Reflection
Normal line (perpendicular to
surface)
inc
ys
i r
ra
ide
t ed
nt
c
fle
ra y
re
s
i=r

4. Diffuse Reflection
Diffuse reflection is when light bounces off a non-smooth surface.
Each ray of light still obeys the law of reflection, but because the
surface is not smooth, the normal can point in a different for
every ray. If many light rays strike a non-smooth surface, they
could be reflected in many different directions. This explains
how we can see objects even when it seems the light shining upon
it should not reflect in the direction of our eyes. It also helps to
explain glare on wet roads: Water fills in and smoothes out the
rough road surface so that the road becomes more like a mirror.

5. Speed of Light & Refraction
As you have already learned, light is extremely fast, about
3 × 108 m/s in a vacuum. Light, however, is slowed down by the
presence of matter. The extent to which this occurs depends on
what the light is traveling through. Light travels at about 3/4 of its
vacuum speed (0.75 c ) in water and about 2/3 its vacuum speed
(0.67 c ) in glass. The reason for this slowing is because when
light strikes an atom it must interact with its electron cloud. If
light travels from one medium to another, and if the speeds in
these media differ, then light is subject to refraction (a changing
of direction at the interface).
Refraction of Refraction of
light waves light rays

6. Reflection & Refraction
At an interface between two media, both reflection and refraction can
occur. The angles of incidence, reflection, and refraction are all measured
with respect to the normal. The angles of incidence and reflection are
always the same. If light speeds up upon entering a new medium, the angle
of refraction, θr , will be greater than the angle of incidence, as depicted on
the left. If the light slows down in the new medium, θr will be less than
the angle of incidence, as shown on the right.
Inc Ray Inc ay
ide te d ide dR
nt lec nt c te
Ra ef Ra fle
y R y Re
θr
Re
Refr
normal
normal
ac
fr
ted R
act
ay
θr
ed
R
ay

7. Axle Analogy
Imagine you’re on a skateboard heading from the sidewalk toward some
grass at an angle. Your front axle is depicted before and after entering the
grass. Your right contacts the grass first and slows, but your left wheel is
still moving quickly on the sidewalk. This causes a turn toward the normal.
If you skated from grass to sidewalk, the same path would be followed. In
this case your right wheel would reach the sidewalk first and speed up, but
your left wheel would still be moving more slowly. The result this time
would be turning away from the normal. Skating from sidewalk to grass is
like light traveling from air to a more
overhead view
“optically dense” medium like glass
or water. The slower light travels in
the new medium, the more it bends
toward the normal. Light traveling
sidewalk
from water to air speeds up and grass
bends away from the normal. As
with a skateboard, light traveling
along the normal will change speed θr
but not direction.

8. Index of Refraction, n
The index of refraction of a substance is the ratio of the speed in light
in a vacuum to the speed of light in that substance:
c
n=
v
Medium n
n = Index of Refraction
Vacuum 1
c = Speed of light in vacuum
Air (STP) 1.00029
v = Speed of light in medium
Water (20º C) 1.33
Note that a large index of refraction
corresponds to a relatively slow Ethanol 1.36
light speed in that medium. Glass ~1.5
Diamond 2.42

9. θi
Snell’s Law ni
nr
θr
Snell’s law states that a ray of light bends in
such a way that the ratio of the sine of the
angle of incidence to the sine of the angle of
refraction is constant. Mathematically,
ni sinθ i = nr sinθr
Here ni is the index of refraction in the original
medium and nr is the index in the medium the
light enters. θ i and θr are the angles of
incidence and refraction, respectively.
Willebrord
Snell

10. Snell’s Law Derivation Two parallel rays are shown.
Points A and B are directly
opposite one another. The top
pair is at one point in time, and
the bottom pair after time t.
A θ1 The dashed lines connecting
n1 x • the pairs are perpendicular to
A the rays. In time t, point A
d
• •B travels a distance x, while
y
n2 point B travels a distance y.
•B
sinθ1 = x / d, so x = d sinθ1
θ2 sinθ2 = y / d, so y = d sinθ2
Speed of A: v1 = x / t
Speed of B: v2 = y / t
Continued…

11. Snell’s Law Derivation
(cont.)
A θ1
n1 x •
A d
• •B v1 x/ t x sinθ1
y = = = So,
n2 •B v2 y/ t y sinθ2
θ2
v1 / c sinθ1 1 / n1 sinθ1 n2
= ⇒ = =
v2 / c sinθ2 1 / n2 sinθ2 n1
⇒ n1 sinθ1 = n2 sinθ2

12. Refraction Problem #1
Goal: Find the angular displacement of the ray after having passed
through the prism. Hints: 1. Find the first angle of refraction
using Snell’s law. 19.4712º
2. Find angle ø. (Hint: Use
Geometry skills.) 79.4712º
Air, n1 = 1
30° 3. Find the second angle of
incidence. 10.5288º
4. Find the second angle of
Horiz. ray, refraction, θ, using Snell’s Law
parallel to
ø
θ 15.9º
base
Glass, n2 = 1.5

13. Refraction Problem #2
Goal: Find the distance the light ray displaced due to the thick
window and how much time it spends in the glass. Some hints are
given.
20º θ1 1. Find θ1 (just for fun). 20º
H20
n1 = 1.3 2. To show incoming & outgoing
rays are parallel, find θ. 20º
3. Find d. 0.504 m
glass
10m
n2 = 1.5 4. Find the time the light spends in
5.2 · 10-8 s
d H20 the glass.
θ
Extra practice: Find θ if bottom
medium is replaced with air.
26.4º

14. Refraction Problem #3
Goal: Find the exit angle relative to the horizontal.
θ = 19.8°
36°
air
glass θ=?
The triangle is isosceles.
Incident ray is horizontal,
parallel to the base.

15. Reflection Problem
Goal: Find incident angle relative to horizontal so that reflected ray
will be vertical.
θ = 10º
θ
50º
center of
semicircular mirror
with horizontal base

16. Brewster Angle
The Brewster angle is the angle of incidence the produces reflected
and refracted rays that are perpendicular.
From Snell, n1 sinθb = n2 sinθ.
n2 θ
α = θb since α + β = 90º, α
and θb + β = 90º. β
n1
θb θb
β = θ since α + β = 90º,
and θ + α = 90º. Thus,
n1 sinθb = n2 sinθ = n2 sinβ = n2 cosθb
tanθb = n2 / n1 Sir David
Brewster

17. Critical Angle
The incident angle that causes nr
the refracted ray to skim right
ni
along the boundary of a θc
substance is known as the critical
angle, θc. The critical angle is the
angle of incidence that produces From Snell,
an angle of refraction of 90º. If n1 sinθc = n2 sin 90°
the angle of incidence exceeds
the critical angle, the ray is Since sin 90° = 1, we
completely reflected and does have n1 sinθc = n2 and
not enter the new medium. A the critical angle is
critical angle only exists when
light is attempting to penetrate a nr
medium of higher optical density θc = sin-1
than it is currently traveling in.
ni

18. Critical Angle Sample Problem
Calculate the critical angle for the diamond-air boundary.
Refer to the Index of Refraction chart for the information.
air θc = sin-1 (nr / ni)
diamond = sin-1 (1 / 2.42)
θc
= 24.4°
Any light shone on this
boundary beyond this angle
will be reflected back into the
diamond.

19. Total Internal Reflection
Total internal reflection occurs when light attempts to pass
from a more optically dense medium to a less optically dense
medium at an angle greater than the critical angle. When this
occurs there is no refraction, only reflection.
n1 n2 > n1
n2 θ θ > θc
Total internal reflection can be used for practical applications
like fiber optics.

20. Fiber Optics
Fiber optic lines are strands of glass or
transparent fibers that allows the transmission
of light and digital information over long
distances. They are used for the telephone
system, the cable TV system, the internet,
medical imaging, and mechanical engineering
spool of optical fiber inspection.
Optical fibers have many advantages over
copper wires. They are less expensive,
thinner, lightweight, and more flexible. They
aren’t flammable since they use light signals
instead of electric signals. Light signals from
one fiber do not interfere with signals in
nearby fibers, which means clearer TV A fiber optic wire
reception or phone conversations.
Continued…

21. Fiber Optics Cont.
Fiber optics are often long strands
of very pure glass. They are very
thin, about the size of a human
hair. Hundreds to thousands of
them are arranged in bundles
(optical cables) that can transmit
light great distances. There are
three main parts to an optical
fiber:
• Core- the thin glass center where light travels.
• Cladding- optical material (with a lower index of refraction
than the core) that surrounds the core that reflects light back into
the core.
• Buffer Coating- plastic coating on the outside of an optical
fiber to protect it from damage. Continued…

22. Light travels through the core of a
fiber optic by continually Fiber Optics (cont.)
reflecting off of the cladding. Due
to total internal reflection, the
cladding does not absorb any of
the light, allowing the light to There are two types of optical
travel over great distances. Some fibers:
of the light signal will degrade • Single-mode fibers- transmit
over time due to impurities in the one signal per fiber (used in
glass. cable TV and telephones).
• Multi-mode fibers- transmit
multiple signals per fiber (used
in computer networks).

23. Mirage Pictures

24. Mirages
Mirages are caused by the refracting properties of a
non-uniform atmosphere.
Several examples of mirages include seeing “puddles”
ahead on a hot highway or in a desert and the lingering
daylight after the sun is below the horizon.
More Mirages
Continued…

25. Inferior Mirages
A person sees a puddle ahead on
the hot highway because the road
heats the air above it, while the
air farther above the road stays
cool. Instead of just two layers,
hot and cool, there are really
many layers, each slightly hotter than the layer above it. The cooler air has a
slightly higher index of refraction than the warm air beneath it. Rays of
light coming toward the road gradually refract further from the normal,
more parallel to the road. (Imagine the wheels and axle: on a light ray
coming from the sky, the left wheel is always in slightly warmer air than the
right wheel, so the left wheel continually moves faster, bending the axle
more and more toward the observer.) When a ray is bent enough, it
surpasses the critical angle and reflects. The ray continues to refract as it
heads toward the observer. The “puddle” is really just an inverted image of
the sky above. This is an example of an inferior mirage, since the cool are is
above the hot air.

26. Superior Mirages
Superior mirages occur when a
layer of cool air is beneath a layer
of warm air. Light rays are bent
downward, which can make an
object seem to be higher in the air
and inverted. (Imagine the
wheels and axle on a ray coming
from the boat: the right wheel is
continually in slightly warmer air
than the left wheel. Thus, the right
wheel moves slightly faster and
bends the axle toward the
observer.) When the critical angle
is exceeded the ray reflects. These
mirages usually occur over ice, snow, or cold water. Sometimes superior
images are produced without reflection. Eric the Red, for example, was able to
see Greenland while it was below the horizon due to the light gradually
refracting and following the curvature of the Earth.

27. Sunlight after Sunset
Lingering daylight after the sun
is below the horizon is another Apparent
effect of refraction. Light travels position Observer
at a slightly slower speed in of sun
Earth’s atmosphere than in
space. As a result, sunlight is
Actual
refracted by the atmosphere. In
position Earth
the morning, this refraction
of sun
causes sunlight to reach us
before the sun is actually above Atmosphere
the horizon. In the evening, the
sunlight is bent above the horizon after the sun has actually set. So
daylight is extended in the morning and evening because of the
refraction of light. Note: the picture greatly exaggerates this effect as
well as the thickness of the atmosphere.
Different “shapes” of Sun

28. Dispersion of Light
Dispersion is the separation of light into a spectrum by refraction. The
index of refraction is actually a function of wavelength. For longer
wavelengths the index is slightly small. Thus, red light refracts less than
violet. (The pic is exaggerated.) This effect causes white light to split
into it spectrum of colors. Red light travels the fastest in glass, has a
smaller index of refraction, and bends the least. Violet is slowed down
the most, has the largest index, and bends the most. In other words: the
higher the frequency, the greater the bending.
Animation

29. Atmospheric Optics
There are many natural occurrences of light optics in our atmosphere.
One of the most common of these is
the rainbow, which is caused by
water droplets dispersing sunlight.
Others include arcs, halos, cloud
iridescence, and many more.
Photo gallery of atmospheric optics.

30. Rainbows A rainbow is a spectrum
formed when sunlight is
dispersed by water droplets in
the atmosphere. Sunlight
incident on a water droplet is
refracted. Because of
dispersion, each color is
refracted at a slightly different
angle. At the back surface of
the droplet, the light undergoes
total internal reflection. On the
way out of the droplet, the light is once more refracted and dispersed.
Although each droplet produces a complete spectrum, an observer will
only see a certain wavelength of light from each droplet. (The wavelength
depends on the relative positions of the sun, droplet, and observer.)
Because there are millions of droplets in the sky, a complete spectrum is
seen. The droplets reflecting red light make an angle of 42 o with respect to
the direction of the sun’s rays; the droplets reflecting violet light make an
angle of 40o. Rainbow images

31. Primary Rainbow

32. Secondary Secondary Rainbow
The secondary rainbow is a rainbow of radius
51°, occasionally visible outside the primary
rainbow. It is produced when the light
Primary
entering a cloud droplet is reflected twice
internally and then exits the droplet. The color
spectrum is reversed in respect to the primary
rainbow, with red appearing on its inner edge.
Alexander’s
dark region

33. Supernumerary Arcs
Supernumerary arcs are faint arcs of color
just inside the primary rainbow. They
occur when the drops are of uniform size.
If two light rays in a raindrop are
scattered in the same direction but have
take different paths within the drop, then
they could interfere with each other
constructively or destructively. The type
of interference that occurs depends on the
difference in distance traveled by the
rays. If that difference is nearly zero or a
multiple of the wavelength, it is
constructive, and that color is reinforced.
If the difference is close to half a
wavelength, there is destructive
interference.

34. Real vs. Virtual Images
Real images are formed by mirrors or lenses when light rays
actually converge and pass through the image. Real images will be
located in front of the mirror forming them. A real image can be
projected onto a piece of paper or a screen. If photographic film
were placed here, a photo could be created.
Virtual images occur where light rays only appear to have
originated. For example, sometimes rays appear to be coming from
a point behind the mirror. Virtual images can’t be projected on
paper, screens, or film since the light rays do not really converge
there.
Examples are forthcoming.

35. Plane Mirror
Object
Rays emanating from an object at point P
strike the mirror and are reflected with equal
angles of incidence and reflection. After
P P’
reflection, the rays continue to spread. If we
extend the rays backward behind the mirror, Virtual
they will intersect at point P’, which is the Image
image of point P. To an observer, the rays
appear to come from point P’, but no source is
there and no rays actually converging there .
For that reason, this image at P’ is a virtual
image. do di
O I
The image, I, formed by a plane mirror
of an object, O, appears to be a
distance di , behind the mirror, equal
to the object distance do.
Animation Continued…

36. Plane Mirror (cont.)
Two rays from object P strike the mirror at points B and M. Each ray is
reflected such that i = r.
Triangles BPM and BP’M are P do B di P’
congruent by ASA (show this),
which implies that do= di and
h = h’. Thus, the image is the h M h’
same distance behind the mirror
Object Image
as the object is in front of it, and
the image is the same size as the
object.
object image
Mirror
With plane mirrors, the image is reversed left to right (or the front and
back of an image ). When you raise your left hand in front of a mirror,
your image raises its right hand. Why aren’t top and bottom reversed?

37. Concave and Convex Mirrors
Concave and convex mirrors are curved mirrors similar to portions
of a sphere.
light rays light rays
Concave mirrors reflect light Convex mirrors reflect light
from their inner surface, like from their outer surface, like
the inside of a spoon. the outside of a spoon.

38. Concave Mirrors
• Concave mirrors are approximately spherical and have a principal
axis that goes through the center, C, of the imagined sphere and ends
at the point at the center of the mirror, A. The principal axis is
perpendicular to the surface of the mirror at A.
• CA is the radius of the sphere,or the radius
of curvature of the mirror, R .
• Halfway between C and A is the focal
point of the mirror, F. This is the point
where rays parallel to the principal axis will
converge when reflected off the mirror.
• The length of FA is the focal length, f.
• The focal length is half of the radius of the
sphere (proven on next slide).

39. r = 2f
To prove that the radius of curvature of a concave mirror is
twice its focal length, first construct a tangent line at the
point of incidence. The normal is perpendicular to the
tangent and goes through the center, C. Here, i = r = β. By
alt. int. angles the angle at C is also β, and α = 2 β. s is the
arc length from the principle axis to the pt. of incidence.
Now imagine a sphere centered
at F with radius f. If the incident
tan
ge
ray is close to the principle axis,
β
ntl
the arc length of the new sphere β s
ine
is about the same as s. From β α
s = r θ, we have s = r β and •
C • f
F
s ≈ f α = 2 f β. Thus, r β ≈ 2 f β,
and r = 2 f. r

40. Focusing Light with Concave Mirrors
Light rays parallel to the principal axis will be
reflected through the focus (disregarding spherical
aberration, explained on next slide.)
In reverse, light rays passing through the
focus will be reflected parallel to the
principal axis, as in a flood light.
Concave mirrors can form both real and virtual images, depending on
where the object is located, as will be shown in upcoming slides.

41. Spherical Aberration
F
•
F
•
C •
C
•
Spherical Mirror Parabolic Mirror
Only parallel rays close to the principal axis of a spherical mirror will
converge at the focal point. Rays farther away will converge at a point
closer to the mirror. The image formed by a large spherical mirror will be
a disk, not a point. This is known as spherical aberration.
Parabolic mirrors don’t have spherical aberration. They are used to focus
rays from stars in a telescope. They can also be used in flashlights and
headlights since a light source placed at their focal point will reflect light
in parallel beams. However, perfectly parabolic mirrors are hard to make
and slight errors could lead to spherical aberration. Continued…

42. Spherical vs. Parabolic Mirrors
Parallel rays converge at the Parabolic mirrors have no
focal point of a spherical spherical aberration. The
mirror only if they are close to mirror focuses all parallel rays
the principal axis. The image at the focal point. That is why
formed in a large spherical they are used in telescopes and
mirror is a disk, not a point light beams like flashlights and
(spherical aberration). car headlights.

43. Concave Mirrors: Object beyond C
object The image formed
when an object is
placed beyond C is
•
C
•
F located between C and
F. It is a real, inverted
image
image that is smaller in
size than the object.
Animation 1
Animation 2

44. Concave Mirrors: Object between C and F
The image formed
object when an object is
placed between C and F
•
C
•
F is located beyond C. It
is a real, inverted image
image that is larger in size
than the object.
Animation 1
Animation 2

45. Concave Mirrors: Object in front of F
The image formed
when an object is
placed in front of F is
object located behind the
image
mirror. It is a virtual,
•
C
•
F upright image that is
larger in size than the
object. It is virtual
since it is formed only
Animation where light rays seem
to be diverging from.

46. Concave Mirrors: Object at C or F
What happens when an object is placed at C?
The image will be formed at C also, but it
will be inverted. It will be real and the
same size as the object.
Animation
What happens when an object is placed at F?
No image will be formed. All rays will
reflect parallel to the principal axis and will
never converge. The image is “at infinity.”

47. Convex Mirrors
• A convex mirror has the
same basic properties as a light rays
concave mirror but its focus
and center are located behind
the mirror.
• This means a convex mirror
has a negative focal length • Rays parallel to the principal
(used later in the mirror axis will reflect as if coming
equation). from the focus behind the
mirror.
• Light rays reflected from
convex mirrors always • Rays approaching the mirror
diverge, so only virtual on a path toward F will reflect
images will be formed. parallel to the principal axis.

48. Convex Mirror Diagram
The image formed by
a convex mirror no
matter where the
object object is placed will
image
be virtual, upright,
•
F
•
C
and smaller than the
object. As the object
is moved closer to the
mirror, the image will
approach the size of
the object.

49. Mirror/Lens Equation Derivation
From ∆PCO, β = θ + α, so 2β = 2θ + 2α.
From ∆PCO, γ = 2θ + α , so -γ = -2θ - α.
P Adding equations yields 2β - γ = α.
θ object From s = r θ, we have
s θ
γ s = r β, s ≈ di α, and
β α
T •
C O s ≈ di α (for rays
image close to the principle
axis). Thus:
s α≈ s
β= r d
o
di
γ≈ s
do di
(cont.)

50. Mirror/Lens Equation Derivation (cont.)
From the last slide, β = s / r, α ≈ s / d0 , γ ≈ s / di , and 2 β - γ = α.
Substituting into the last equation yields:
P
2s s s
s θ
θ object
r -d = d
i o
γ β α 2 1 1
T •
C O r = di + do
image
2 1 1
= d +d
2f i o
di 1 1 1
= d +d
f i o
do
The last equation applies to convex and concave mirrors, as well as to
lenses, provided a sign convention is adhered to.

51. Mirror Sign Convention
f = focal length
1 1 1
di = image distance
f = d i + do
do = object distance
+ for real image
di
- for virtual image
+ for concave mirrors
f
- for convex mirrors

52. Magnification
hi
By definition, m =
ho
m = magnification
hi = image height (negative means inverted)
ho = object height
Magnification is simply the ratio of image height
to object height. A positive magnification means
an upright image.

53. hi -di
Magnification Identity: m = =
ho do
To derive this let’s look at two rays. One hits the mirror on the axis.
The incident and reflected rays each make angle θ relative to the axis.
A second ray is drawn through the center and is reflected back on top
of itself (since a radius is always perpendicular to an tangent line of a
circle). The intersection of
the reflected rays
object
determines the location of
θ ho the tip of the image. Our
• C
result follows
image, from similar triangles, with
the negative sign a
height = hi
consequence of our sign
convention. (In this picture
di do hi is negative and di is
positive.)

54. Mirror Equation Sample Problem
Suppose AllStar, who is 3 and
a half feet tall, stands 27 feet
in front of a concave mirror
with a radius of curvature of
•
C
•
F 20 feet. Where will his image
be reflected and what will its
size be?
di = 15.88 feet
hi = -2.06 feet

55. Mirror Equation Sample Problem 2
Casey decides to join in
the fun and she finds a
convex mirror to stand
in front of. She sees her
image reflected 7 feet
behind the mirror which
•
F
•
C has a focal length of 11
feet. Her image is 1
foot tall. Where is she
standing and how tall is
she? d =19.25 feet
o
ho = 2.75 feet

56. Lenses
Lenses are made of transparent Convex (Converging)
materials, like glass or plastic, that Lens
typically have an index of refraction
greater than that of air. Each of a lens’
two faces is part of a sphere and can be
convex or concave (or one face may be
flat). If a lens is thicker at the center
than the edges, it is a convex, or Concave (Diverging)
converging, lens since parallel rays will Lens
be converged to meet at the focus. A
lens which is thinner in the center than
the edges is a concave, or diverging,
lens since rays going through it will be
spread out.

57. Lenses: Focal Length
• Like mirrors, lenses have a principal axis perpendicular to their
surface and passing through their midpoint.
• Lenses also have a vertical axis, or principal plane, through their
middle.
• They have a focal point, F, and the focal length is the distance from
the vertical axis to F.
• There is no real center of curvature, so 2F is used to denote twice
the focal length.

58. Ray Diagrams For Lenses
When light rays travel through a lens, they refract at both surfaces of
the lens, upon entering and upon leaving the lens. At each interface
the bends toward the normal. (Imagine the wheels and axle.) To
simplify ray diagrams, we often pretend that all refraction occurs at
the vertical axis. This simplification works well for thin lenses and
provides the same results as refracting the light rays twice.
• •
2F F • 2F
F • • •
2F F • 2F
F •
Reality Approximation

59. Convex Lenses
Rays traveling parallel to the principal
axis of a convex lens will refract toward • •
2F F • 2F
F •
the focus.
Rays traveling from the focus will
• F
2F • • 2F
F • refract parallel to the principal axis.
Rays traveling directly through the
center of a convex lens will leave the • •
2F F • 2F
F •
lens traveling in the exact same
direction.

60. Convex Lens: Object Beyond 2F
The image formed
when an object is
object
placed beyond 2F
is located behind
• • • • the lens between F
2F F F 2F and 2F. It is a real,
image inverted image
which is smaller
than the object
Experiment with this diagram
itself.

61. Convex Lens: Object Between 2F and F
The image formed
object when an object is
placed between
2F and F is
•
2F
•
F
•
F
•
2F located beyond 2F
behind the lens.
It is a real,
image inverted image,
larger than the
object.

62. Convex Lens: Object within F
The image formed when an
object is placed in front of
F is located somewhere
image beyond F on the same side
of the lens as the object. It
is a virtual, upright image
•
2F
•
F
•
F
•
2F which is larger than the
object object. This is how a
magnifying glass works.
When the object is brought
close to the lens, it will be
convex lens used magnified greatly.
as a magnifier

63. Concave Lenses
Rays traveling parallel to the
principal axis of a concave lens will
•
2 •
F •
F •
2
refract as if coming from the focus.
F F
Rays traveling toward the
2F •
• F • 2
F •
focus will refract parallel to
the principal axis.
F
Rays traveling directly through the
2F •
• F • 2
F • center of a concave lens will leave
the lens traveling in the exact same
F
direction, just as with a convex lens.

64. Concave Lens Diagram
No matter where the
object is placed, the
object
image will be on the
same side as the
•
2F
•
F
•
F
•
2F object. The image is
image virtual, upright, and
smaller than the object
with a concave lens.
Experiment with this diagram

65. Lens Sign Convention
f = focal length
1 1 1
f = d +d di = image distance
i o
do = object distance
+ for real image
di
- for virtual image
+ for convex lenses
f
- for concave lenses

66. Lens / Mirror Sign Convention
The general rule for lenses and mirrors is this:
+ for real image
di
- for virtual image
and if the lens or mirror has the ability to converge light,
f is positive. Otherwise, f must be treated as negative for
the mirror/lens equation to work correctly.

67. Lens Sample Problem
Tooter, who stands 4 feet
tall (counting his
snorkel), finds himself 24
feet in front of a convex
lens and he sees his
image reflected 35 feet
•
2F
•
F
•
F
•
2F behind the lens. What is
the focal length of the
lens and how tall is his
image?
f = 14.24 feet
hi = -5.83 feet

68. Lens and Mirror Applet
This application shows where images will be formed
with concave and convex mirrors and lenses. You can
change between lenses and mirrors at the top. Changing
the focal length to negative will change between
concave and convex lenses and mirrors. You can also
move the object or the lens/mirror by clicking and
dragging on them. If you click with the right mouse
button, the object will move with the mirror/lens. The
focal length can be changed by clicking and dragging at
the top or bottom of the lens/mirror. Object distance,
image distance, focal length, and magnification can also
be changed by typing in values at the top.
Lens and Mirror Diagrams

69. Convex Lens in Water
Glass Glass
H2O Air
Because glass has a higher index of refraction that water the convex
lens at the left will still converge light, but it will converge at a
greater distance from the lens that it normally would in air. This is
due to the fact that the difference in index of refraction between
water and glass is small compared to that of air and glass. A large
difference in index of refraction means a greater change in speed of
light at the interface and, hence, a more dramatic change of
direction.

70. Convex Lens Made of Water
Glass
Since water has a higher index of
refraction than air, a convex lens made of
water will converge light just as a glass
lens of the same shape. However, the
Air glass lens will have a smaller focal length
n = 1.5 than the water lens (provided the lenses
are of same shape) because glass has an
index of refraction greater than that of
water. Since there is a bigger difference
H2O in refractive index at the air-glass
interface than at the air-water interface,
the glass lens will bend light more than
the water lens.
Air
n = 1.33

71. Air & Water Lenses
On the left is depicted a concave lens filled
with water, and light rays entering it from an
air-filled environment. Water has a higher
index than air, so the rays diverge just like
Air they do with a glass lens.
Concave lens made of H2O
To the right is an air-filled convex lens
submerged in water. Instead of
converging the light, the rays diverge
because air has a lower index than water. H2O
Convex lens made of Air
What would be the situation with a concave lens made of air
submerged in water?

72. Chromatic Aberration
As in a raindrop or a prism, different wave-
lengths of light are refracted at different
angles (higher frequency ↔ greater
bending). The light passing through a lens
is slightly dispersed, so objects viewed
through lenses will be ringed with color.
This is known as chromatic aberration and
it will always be present when a single lens Chromatic Aberration
is used. Chromatic aberration can be
greatly reduced when a convex lens is
combined with a concave lens with a
different index of refraction. The
dispersion caused by the convex lens will
be almost canceled by the dispersion
caused by the concave lens. Lenses such as
this are called achromatic lenses and are Achromatic Lens
used in all precision optical instruments.
Examples

73. Human eye
The human eye is a fluid-filled object that
focuses images of objects on the retina. The
cornea, with an index of refraction of about
1.38, is where most of the refraction occurs.
Some of this light will then passes through
the pupil opening into the lens, with an index
of refraction of about 1.44. The lens is flexi- Human eye w/rays
ble and the ciliary muscles contract or relax to change its shape and
focal length. When the muscles relax, the lens flattens and the focal
length becomes longer so that distant objects can be focused on the
retina. When the muscles contract, the lens is pushed into a more
convex shape and the focal length is shortened so that close objects
can be focused on the retina. The retina contains rods and cones to
detect the intensity and frequency of the light and send impulses to the
brain along the optic nerve.

74. Hyperopia The first eye shown suffers from
farsightedness, which is also known
as hyperopia. This is due to a focal
length that is too long, causing the
image to be focused behind the retina,
making it difficult for the person to
see close up things.
Formation of image behind The second eye is being helped with a
the retina in a hyperopic eye. convex lens. The convex lens helps
the eye refract the light and decrease
the image distance so it is once again
focused on the retina.
Hyperopia usually occurs among
adults due to weakened ciliary
Convex lens correction muscles or decreased lens flexibility.
for hyperopic eye.
Farsighted means “can see far” and the rays focus too far from the lens.

75. The first eye suffers from
Myopia nearsightedness, or myopia. This is
a result of a focal length that is too
short, causing the images of distant
objects to be focused in front of the
retina.
The second eye’s vision is being
Formation of image in front corrected with a concave lens. The
of the retina in a myopic eye. concave lens diverges the light rays,
increasing the image distance so that
it is focused on the retina.
Nearsightedness is common among
young people, sometimes the result
of a bulging cornea (which will
Concave lens correction refract light more than normal) or an
for myopic eye. elongated eyeball.
Nearsighted means “can see near” and the rays focus too near the lens.

76. Refracting Telescopes
Refracting telescopes are comprised of two convex lenses. The objective
lens collects light from a distant source, converging it to a focus and
forming a real, inverted image inside the telescope. The objective lens
needs to be fairly large in order to have enough light-gathering power so
that the final image is bright enough to see. An eyepiece lens is situated
beyond this focal point by a distance equal to its own focal length. Thus,
each lens has a focal point at F. The rays exiting the eyepiece are nearly
parallel, resulting in a magnified, inverted, virtual image. Besides
magnification, a good telescope also needs resolving power, which is its
ability to distinguish objects with very small angular separations.
F

77. Reflecting Telescopes
Galileo was the first to use a refracting telescope for astronomy. It is
difficult to make large refracting telescopes, though, because the
objective lens becomes so heavy that it is distorted by its own weight. In
1668 Newton invented a reflecting telescope. Instead of an objective
lens, it uses a concave objective mirror, which focuses incoming parallel
rays. A small plane mirror is placed at this focal point to shoot the light
up to an eyepiece lens (perpendicular to incoming rays) on the side of
the telescope. The mirror serves to gather as much light as possible,
while the eyepiece lens, as in the refracting scope, is responsible for the
magnification.

78. Huygens’ Principle
Christiaan Huygens, a contemporary of Newton, was
an advocate of the wave theory of light. (Newton
favored the particle view.) Huygens’ principle states
that a wave crest can be thought of as a series of
equally-spaced point sources that produce wavelets
that travel at the same speed as the original wave.
These wavelets superimpose with one another.
Constructive interference occurs along a line parallel
to the original wave at a distance of one wavelength
from it. This principle explains diffraction well:
When light passes through a very small slit, it is as if
only one of these point sources is allowed through. Christiaan
Since there are no other sources to interfere with it, Huygens
circular wavefronts radiate outwards in all directions.
Applet showing reflect
• • • • •

79. screen P
Diffraction: Single Slit
Light enters an opening of width a and is
diffracted onto a distant screen. All points at the
opening act as individual point sources of light.
These point sources interfere with each other, both
constructively and destructively, at different points
on the screen, producing alternating bands of
light and dark. To find the first dark spot, let’s
consider two point sources: one at the left edge,
and one in the middle of the slit. Light from the left
point source must travel a greater distance to point
P on the screen than light from the middle point
source. If this extra distance Extra
is a half a wavelength, λ/2, distance
destructive interference will
occur at P and there will
be a dark spot there. a/2
applet a Continued…

80. Single Slit (cont.)
Let’s zoom in on the small triangle in the last slide. Since a / 2 is
extremely small compared to the distanced to the screen, the two
arrows pointing to P are essentially parallel. The extra distance is
found by drawing segment AC perpendicular to BC. This means that
angle A in the triangle is also θ. Since AB is the hypotenuse of a
right triangle, the extra distance is given by (a / 2) sinθ. Thus, using
(a / 2) sinθ = λ/2, or equivalently,
P
a sinθ = λ, we can locate the first dark
i nt
po
C spot on the screen. Other dark spots can
To
be located by dividing the slit further.
e c
an
P
ist
θ θ
int
ad
po
tr
θ
Ex
To
B a/2 A

81. screen P
Diffraction: Double Slit
Light passes through two openings, each
of which acts as a point source. Here a is
the distance between the openings rather
than the width of a particular opening. As
before, if d1 - d2 = n λ (a multiple of the
wavelength), light from the two sources
will be in phase and there will a bright d1
spot at P for that wavelength. By the d2
Pythagorean theorem, the exact difference
L
in distance is
d1 - d2 = [ L2 + (x + a / 2)2 ] ½
- [ L2 + (x - a / 2)2 ] ½
Approximation on next slide.
Link 1 Link 2 a x

82. Double Slit (cont.) screen P
In practice, L is far greater than a, meaning
that segments measuring d1 and d2 are
virtually parallel. Thus, both rays make an
angle θ relative to the vertical, and the
bottom right angle of the triangle is also θ
(just like in the single slit case). This means
the extra distance traveled is given by a sinθ. d1
Therefore, the required condition for a bright d2
spot at P is that there exists a natural number, L
n, such that:
a sinθ = n λ θ θ
If white light is shone at the
slits, different colors will be
in phase at different angles.
Electron diffraction a

83. Diffraction Gratings
A different grating has numerous tiny slits, equally spaced. It
separates white light into its component colors just as a double slit
would. When a sinθ = n λ, light of wavelength λ will be reinforced
at an angle of θ. Since different colors have different wavelengths,
different colors will be reinforced at different angles, and a prism-like
spectrum can be produced. Note, though, that prisms separate light via
refraction rather than diffraction. The pic on the left shows red light
shone through a grating. The CD acts as a diffraction grating since the
tracks are very close together (about 625/mm).

84. Credits
Snork pics: http://www.geocities.com/EnchantedForest/Cottage/7352/indosnor.html
Snorks icons: http://www.iconarchive.com/icon/cartoon/snorks_by_pino/
Snork seahorse pic: http://members.aol.com/discopanth/private/snork.jpg
Mirror, Lens, and Eye pics:
http://www.physicsclassroom.com/
Refracting Telescope pic: http://csep10.phys.utk.edu/astr162/lect/light/refracting.html
Reflecting Telescope pic: http://csep10.phys.utk.edu/astr162/lect/light/reflecting.html
Fiber Optics: http://www.howstuffworks.com/fiber-optic.htm
Willebrord Snell and Christiaan Huygens pics:
http://micro.magnet.fsu.edu/optics/timeline/people/snell.html Chromatic Aberrations:
http://www.dpreview.com/learn/Glossary/Optical/Chromatic_Aberrations_01.htm
Mirage Diagrams:
http://www.islandnet.com/~see/weather/elements/mirage1.htm
Sir David Brewster pic: http://www.brewstersociety.com/brewster_bio.html
Mirage pics: http://www.polarimage.fi/
http://www.greatestplaces.org/mirage/desert1.html
http://www.ac-grenoble.fr/college.ugine/physique/les%20mirages.html
Diffuse reflection: http://www.glenbrook.k12.il.us/gbssci/phys/Class/refln/u13l1d.html
Diffraction: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/grating.html